To simplify the expression \(\frac{x^2 - 10x - 24}{x^2 - 3x - 108}\) and find the restrictions on the variable, we first need to factor both the numerator and the denominator.
Step 1: Factor the numerator \(x^2 - 10x - 24\)
We are looking for two numbers that multiply to \(-24\) (the constant term) and add to \(-10\) (the coefficient of \(x\)). The numbers \(-12\) and \(2\) meet these criteria. Thus, we can factor the numerator as:
\[ x^2 - 10x - 24 = (x - 12)(x + 2) \]
Step 2: Factor the denominator \(x^2 - 3x - 108\)
Next, we need to factor the denominator. We are looking for two numbers that multiply to \(-108\) and add to \(-3\). The numbers \(9\) and \(-12\) meet these criteria. Thus, we can factor the denominator as:
\[ x^2 - 3x - 108 = (x - 12)(x + 9) \]
Step 3: Rewrite the original expression
Now we can rewrite the original expression:
\[ \frac{x^2 - 10x - 24}{x^2 - 3x - 108} = \frac{(x - 12)(x + 2)}{(x - 12)(x + 9)} \]
Step 4: Cancel common factors
We see that \((x - 12)\) is a common factor in both the numerator and the denominator, which we can cancel (with the restriction that \(x \neq 12\)):
\[ \frac{(x + 2)}{(x + 9)}, \quad x \neq 12 \]
Step 5: Determine restrictions on \(x\)
The original expression is undefined where the denominator is zero:
From \(x^2 - 3x - 108\):
\[ x - 12 = 0 \implies x = 12 \] \[ x + 9 = 0 \implies x = -9 \]
Thus, restrictions on \(x\) are \(x \neq 12\) and \(x \neq -9\).
Final Result
The simplified expression is:
\[ \frac{x + 2}{x + 9} \quad \text{for } x \neq 12, x \neq -9 \]
Answer
The correct option is:
\[ \frac{x + 2}{x + 9}, \quad x \neq -9, x \neq 12 \]